This is a generalization of “\(u\)-substitution”. For the integral \(\iint_R f \,\mathrm{d}A\) over a region \(R\) in the \(xy\)-plane, let \({T \colon S \to R}\) be a transformation from the \(uv\)-plane to the \(xy\)-plane — so \({T(u,v) = (x,y)}\) and \({T(R) = S}\) — that’s invertible on the interior of \(S\) and has continuous first-order partial derivatives. The Jacobian matrix \(\mathbf{J}_T\) of \(T\) is the matrix of partial derivatives of \(x\) and \(y\) with respect to \(u\) and \(v,\) and the Jacobian determinant \(\operatorname{J}_T,\) often just called the Jacobian, is the determinant of this matrix and serves as an “area correction factor” by which the integral over \(S\) in the \(uv\)-plane that transforms into \(\iint_R f \,\mathrm{d}A\) via the transformation \(T\) must be scaled to make them equivalent. \[ \mathbf{J}_T = \begin{pmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\[2pt] \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{pmatrix} \quad\implies\quad {\color{maroon}\operatorname{J}_T} = \operatorname{det}\bigl(\mathbf{J}_T\bigr) %= \frac{ \partial(x,y) }{ \partial(u,v) } %= \operatorname{det}\begin{pmatrix} % \frac{\partial x}{\partial u} % & \frac{\partial x}{\partial v} % \\[2pt] \frac{\partial y}{\partial u} % & \frac{\partial y}{\partial v} %\end{pmatrix} = \frac{\partial x}{\partial u} \frac{\partial y}{\partial v} - \frac{\partial x}{\partial v} \frac{\partial y}{\partial u} \qquad \qquad \iint_S f\bigl(x(u,v), y(u,v)\bigr) \,{\color{maroon} \operatorname{J}_T} \,\mathrm{d}u\,\mathrm{d}v \;\;= \;\; \iint_R f \,\mathrm{d}A \] In three-dimensional space, for a transformation \(T \colon \mathbf{R}^3 \to \mathbf{R}^3\) the Jacobian matrix is a \(3\times 3\) matrix but the core idea is the same. The transformations from polar/cylindrical coordinates \((z, r, \theta)\) or spherical coordinates \((\rho, \theta, \varphi)\) into rectangular coordinates \((x,y,z)\) are a particular examples of this, where in the former cases \({\color{maroon} \operatorname{J}_T = r}\) and in the latter case \({\color{maroon} \operatorname{J}_T = \rho^2\sin(\varphi)}.\) \[ \mathbf{J}_T %= \frac{ \partial(x,y,z) }{ \partial(u,v,w) } = \begin{pmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\[2pt] \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\[2pt] \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \end{pmatrix} \quad\implies\quad {\color{maroon}\operatorname{J}_T} = \operatorname{det}\bigl(\mathbf{J}_T\bigr) = \frac{\partial x}{\partial u} \frac{\partial y}{\partial v} \frac{\partial z}{\partial w} - \frac{\partial x}{\partial u} \frac{\partial y}{\partial w} \frac{\partial z}{\partial v} + \frac{\partial x}{\partial v} \frac{\partial y}{\partial w} \frac{\partial z}{\partial u} - \frac{\partial x}{\partial v} \frac{\partial y}{\partial u} \frac{\partial z}{\partial w} + \frac{\partial x}{\partial w} \frac{\partial y}{\partial u} \frac{\partial z}{\partial v} - \frac{\partial x}{\partial w} \frac{\partial y}{\partial v} \frac{\partial z}{\partial u} \\ \iiint_S f\bigl(x(u,v,w), y(u,v,w), z(u,v,w)\bigr) \,{\color{maroon} \operatorname{J}_T} \,\mathrm{d}u\,\mathrm{d}v\,\mathrm{d}w \;\;=\;\; \iiint_E f \,\mathrm{d}V \]
The Jacobian conjecture (1939, unproven) — “For a transformation \(T\colon \mathbf{R}^n \to \mathbf{R}^n\) with polynomial components, if \(\operatorname{J}_T\) is non-zero and constant, then \(T\) is invertible and its inverse transformation \(T^{-1}\colon \mathbf{R}^n \to \mathbf{R}^n\) is also defined by polynomial components.”